
3.3: Systems of First Order

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Consider the quasilinear system

\label{syst1}
\sum_{k=1}^nA^k(x,u)u_{u_k}+b(x,u)=0,

where $$A^k$$ are $$m\times m$$-matrices, sufficiently regular with respect to their arguments, and

$$u=\left(\begin{array}{c} u_1\\ \vdots\\u_m \end{array}\right),\ \ u_{x_k}=\left(\begin{array}{c} u_{1,x_k}\\ \vdots\\u_{m,x_k} \end{array}\right),\ \ b=\left(\begin{array}{c} b_1\\ \vdots\\b_m \end{array}\right).$$

We ask the same question as above: can we calculate all derivatives of $$u$$ in a neighborhood of a given hypersurface $$\mathcal{S}$$ in $$\mathbb{R}$$ defined by $$\chi(x)=0$$, $$\nabla\chi\not=0$$, provided $$u(x)$$ is given on $$\mathcal{S}$$?

For an answer we map $$\mathcal{S}$$ onto a flat surface $$\mathcal{S}_0$$  by using the mapping $$\lambda=\lambda(x)$$ of Section 3.1 and write equation (\ref{syst1}) in new coordinates. Set $$v(\lambda)=u(x(\lambda))$$, then

$$\sum_{k=1}^nA^k(x,u)\chi_{x_k}v_{\lambda_n}=\mbox{terms known on}\ \mathcal{S}_0.$$

We can solve this system with respect to $$v_{\lambda_n}$$, provided that

$$\det\left(\sum_{k=1}^nA^k(x,u)\chi_{x_k}\right)\not=0$$

on $$\mathcal{S}$$.

Definition. Equation

$$\det\left(\sum_{k=1}^nA^k(x,u)\chi_{x_k}\right)=0$$

is called characteristic equation associated to equation (\ref{syst1}) and a surface $${\mathcal{S}}$$: $$\chi(x)=0$$, defined by a solution $$\chi$$, $$\nabla\chi\not=0$$, of this characteristic equation is said to be characteristic surface.

Set

$$C(x,u,\zeta)=\det\left(\sum_{k=1}^nA^k(x,u)\zeta_k\right)$$

for $$\zeta_k\in\mathbb{R}$$.

Definition.

1. The system (\ref{syst1}) is hyperbolic at $$(x,u(x))$$ if there is a regular linear mapping $$\zeta=Q\eta$$, where $$\eta=(\eta_1,\ldots,\eta_{n-1},\kappa)$$, such that there exists $$m$$ {\it real} roots $$\kappa_k=\kappa_k(x,u(x),\eta_1,\ldots,\eta_{n-1})$$, $$k=1,\ldots,m$$, of $$D(x,u(x),\eta_1,\ldots,\eta_{n-1},\kappa)=0$$ for all $$(\eta_1,\ldots,\eta_{n-1})$$, where $$D(x,u(x),\eta_1,\ldots,\eta_{n-1},\kappa)=C(x,u(x),x,Q\eta).$$
2. System (\ref{syst1}) is parabolic if there exists a regular linear mapping $$\zeta=Q\eta$$ such that $$D$$ is independent of $$\kappa$$, that is, $$D$$ depends on less than $$n$$ parameters.
3. System (\ref{syst1}) is elliptic if $$C(x,u,\zeta)=0$$ only if $$\zeta=0$$.

Remark. In the elliptic case all derivatives of the solution can be calculated from the given data and the given equation.

Contributors

• Integrated by Justin Marshall.